On the limit of some Toeplitz-like determinants

In this article we derive, using standard methods of Toeplitz theory, an asymptotic formula for certain large minors of Toeplitz matrices. D. Bump and P. Diaconis obtained the same asymptotics using representation theory, with an answer having a different form.Comment: LaTeX file, 3 page

On lacunary Toeplitz determinants

By using Riemann--Hilbert problem based techniques, we obtain the asymptotic expansion of lacunary Toeplitz determinants $\det_N\big[ c_{\ell_a-m_b}[f] \big]$ generated by holomorhpic symbols, where $\ell_a=a$ (resp. $m_b=b$) except for a finite subset of indices $a=h_1,\dots, h_n$ (resp. $b=t_1,\dots, t_r$). In addition to the usual Szeg\"{o} asymptotics, our answer involves a determinant of size $n+r$.Comment: 11 page

Toeplitz minors and specializations of skew Schur polynomials

We express minors of Toeplitz matrices of finite and large dimension in terms of symmetric functions. Comparing the resulting expressions with the inverses of some Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skew Schur polynomials.Comment: v2: Added new results on specializations of skew Schur polynomials, abstract and title modified accordingly and references added; v3: final, published version; 18 page

Topological field theory approach to intermediate statistics

Random matrix models provide a phenomenological description of a vast variety of physical phenomena. Prominent examples include the eigenvalue statistics of quantum (chaotic) systems, which are conveniently characterized using the spectral form factor (SFF). Here, we calculate the SFF of unitary matrix ensembles of infinite order with the weight function satisfying the assumptions of Szeg\"o's limit theorem. We then consider a parameter-dependent critical ensemble which has intermediate statistics characteristic of ergodic-to-nonergodic transitions such as the Anderson localization transition. This same ensemble is the matrix model of $U(N)$ Chern-Simons theory on $S^3$, and the SFF of this ensemble is proportional to the HOMFLY invariant of $(2n,2)$-torus links with one component in the fundamental and one in the antifundamental representation. This is one of a large class of ensembles arising from topological field and string theories which exhibit intermediate statistics. Indeed, the absence of a local order parameter suggests that it is natural to characterize ergodic-to-nonergodic transitions using topological tools, such as we have done here.Comment: 41 pages, 3 figures. Changes: corrected typos and affiliations, resized figure 1. Submission to SciPos

Schur Averages in Random Matrix Ensembles

The main focus of this PhD thesis is the study of minors of Toeplitz, Hankel and Toeplitz±Hankel matrices. These can be expressed as matrix models over the classical Lie groups G(N) = U(N); Sp(2N);O(2N);O(2N + 1), with the insertion of irreducible characters associated to each of the groups. In order to approach this topic, we consider matrices generated by formal power series in terms of symmetric functions. We exploit these connections to obtain several relations between the models over the different groups G(N), and to investigate some of their structural properties. We compute explicitly several objects of interest, including a variety of matrix models, evaluations of certain skew Schur polynomials, partition functions and Wilson loops of G(N) Chern-Simons theory on S3, and fermion quantum models with matrix degrees of freedom. We also explore the connection with orthogonal polynomials, and study the large N behaviour of the average of a characteristic polynomial in the Laguerre Unitary Ensemble by means of the associated Riemann-Hilbert problem. We gratefully acknowledge the support of the Fundação para a Ciência e a Tecnologia through its LisMath scholarship PD/BD/113627/2015, which made this work possible

On the limit of some Toeplitz-like determinants

In this article we derive, using standard methods of Toeplitz theory, an asymptotic formula for certain large minors of Toeplitz matrices. D. Bump and P. Diaconis obtained the same asymptotics using representation theory, with an answer having a different form